Update: Dec. 20, 2018 - Valve is no longer selling the Steam Link hardware, but it will continue to update and support the Steam Link app going forward. The beta version of Valve's Steam Link app ...

As it is, Steam Link users can stream their PC games over the same network their host machine is connected to, but as the name suggests, Steam Link Anywhere allows them to stream their games to devices on different networks. You can also connect a controller to your phone if you need more precise inputs for more demanding games - like a racer or Dark Souls or something. Valve explain the system here, and among the perks for developers are anonymized network traffic (protecting servers and clients from DDOS attacks) and potentially lower pings through Valve's networks.

A post from slouken on the Steam Community boards stated that you should be able to stream games to your Steam Link from virtually any computer running Steam.